The report states that the Istanbul giants have reached an agreement with the Polish international and are locked in negotiations with Torino over the transfer fee.
Glik has two years remaining on his contract with Torino but is keen on a move away from the Serie A side.
The Polish international has caught the eye of United’s boss Louis van Gaal following an impressive season.
Glik provided two assists and scored eight goals in 44 appearances for Torino in all competitions last season.
Torino finished the season in 9th place with 54 points in 38 games.
Galatasaray won the league title for a record 20th time last season and booked a place in next seasons Champions League group stages.